Evaluation

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Research in Mathematics at Norwegian Universities

An evaluation

Division for Science

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Research in Mathematics at Norwegian Universities

An evaluation

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The Research Council of Norway P.O.Box 2700 St. Hanshaugen N–0131 OSLO

Telephone: +47 22 03 70 00 Telefax: +47 22 03 70 01 bibliotek@rcn.no www.rcn.no/english

The report can be ordered at:

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Design cover: Design et cetera

Printing: 07 Gruppen/The Research Council of Norway Number of copies: 200

Oslo, March 2012

ISBN 978-82-12-03057-2 (print) ISBN 978-82-12-03058-9 (pdf)

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To the Research Council of Norway

The members of the Evaluation Committee for Research in Mathematics in Norwegian Universities hereby submit the following report.

The views presented in this report are the consensus among the members of the Evaluation Committee. The report represents an agreed account of the assessments and recommendations.

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1 Findings and Recommendations

1. Mathematical sciences research and education is vital to the high-tech and engineering industry of Norway. However, there is evidence that Norway is not investing enough in science and technology, of which the mathematical sciences are an essential part.

2. Research in the mathematical sciences at Norwegian universities remains at a very high, international level. Since the last evaluation in 2002, applied mathematics and statistics have seen positive developments, and are now strong. However, there are also signs of decline in some of the other areas where less work of the very top quality is produced.

3. The mathematical sciences face a serious depletion problem with many retirements coming up but not enough young mathematicians in the 'pipeline'. Many more graduate students and post- docs are needed.

4. In order to develop a clear career structure in academia, the Committee recommends the creation of career development posts that can bridge the gap between a post-doc position and tenure.

5. Women remain seriously under-represented within the mathematical sciences. It is particularly concerning that the proportion of women amongst younger mathematicians is not increasing.

Decisive action is needed to ensure that Norwegian mathematics does not miss many opportunities in future by not being able to attract and support women in their careers.

6. The level of collaboration between mathematicians in different universities is generally good and benefits from the grant support by the RCN. However, this support is not always sufficient.

7. Present programme-driven research initiatives favoured by funding agencies are not always a good fit for the mathematical sciences, especially the more theoretical parts. The RCN is strongly encouraged to enlarge substantially its portfolio for investigator-driven basic research able to support also individuals and small research groups.

8. It is generally considered that mobility enhances research productivity. Tools for every stage of the career should be developed to encourage mobility between Norwegian universities and those abroad, as well as between universities and industry. Academic careers should favour mobility.

9. The mathematical sciences form an infrastructure for other sciences and technology. An

increase in funds for mathematical research and education – both in schools and at universities – is now urgently needed to ensure that also in future the R&D capabilities underpinning Norway's industry are adequately served.

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2 Introduction

2.1 Mandate and the Review Process

This report was commissioned by the Research Council of Norway (RCN) and presents an evaluation of research in Mathematics in Norwegian Universities with particular focus on the period 2006-2010. The detailed terms of reference and the mandate for this review are provided in full in Appendix C.

The main objectives and the process of this review are given here in summary.

Objectives

The primary goals of this report are to:

 Provide a critical review in an international context of the strengths and weaknesses of research in Mathematics in Norwegian universities; at the national level, at the level of departments, and at the level of research groups (but not of individuals).

 Identify research groups that have achieved a high international level of quality or that have the potential to reach such a level.

 Identify areas of research that need to be strengthened in order to ensure that Norway in the future will have the necessary competence in areas of national importance.

 Examine to what extent research in mathematics meets the demands of interdisciplinary research and future societal challenges.

 Assess the degree of national and international mobility; between different research groups as well as between universities and industry.

 Assess to what degree the previous evaluation from 2002¹ has been used by the institutions in their strategic planning.

 For the institutions, provide advice and recommendations that can be used to enhance their own research standards and strategies.

 For the RCN, provide an improved knowledge base that can be used by the RCN in its role as a funder of research in Mathematics and as an adviser on research policies to the Norwegian Government.

Process

The Committee’s evaluation was carried out on the basis of:

 Fact sheets with information about the departments and research groups.

 Self-evaluations provided by the departments and research groups.

 Assessment of publications and citations.²

 Hearings conducted between the Evaluation Committee and the participating departments and research groups.

All fact sheets, a description and the schedule of the hearings can be found in Appendix D.

1 Evaluation: Research in Mathematics in Norwegian Universities and Colleges. The Research Council of Norway, 2002.

2 Including NIFU: Evaluation of Mathematics – Publication and Citation Analysis, Dag W. Aksnes. However, the report was delayed and the Committee did not have the opportunity to examine systematically the NIFU report in the light of its own findings, or vice versa.

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2.2 Participants of the Evaluation

The relevant departments and other research bodies were invited to participate in this evaluation by the RCN in October 2010. All of the university colleges and contract research institutes contacted opted out.

While some of these research units had already been included in other evaluations, others did not respond. It should be noted that there are more than 100 mathematicians working at these places and not included in this report.

The participating institutions are the following eight departments at seven different universities:

Department of Mathematics and Natural Sciences at the University of Stavanger Department of Mathematical Sciences at the University of Adger

Department of Mathematics at the University of Oslo Department of Mathematics at the University of Bergen

Department of Mathematical Sciences at the Norwegian University of Science and Technology (NTNU)

Department of Mathematics and Statistics at the University of Tromsø Norwegian University of Life Sciences:

- Department of Mathematical Sciences and Technology - Department of Chemistry, Biotechnology and Food Chemistry.

2.3 Key Figures

The Committee was provided with fact sheets, included in Appendix D of this report, giving an overview of basic factual information on the departments and research groups. These data were collected by all participating institutions simultaneously with a deadline in May 2011. They are taken to represent a fair and unbiased snapshot of all participating departments and groups at the time. All staffing and graduate numbers quoted in this report are taken from these fact sheets. It should be noted that these numbers sometimes differ from those presented in the self-evaluations or, several months later, during the hearings.

2.4 Previous Evaluation

The previous evaluation of research in Mathematics in Norway was carried out at the request of the RCN in 2001-2002. It led to a set of recommendations, which have been taken into consideration during the present evaluation, and which are commented on in the context of some of the participating institutions in this report.

2.5 Grading

For the assessment of the research groups, a grading system has been applied that, in keeping with the mandate, focuses on the following aspects:

 Quality of international publications

 Level of productivity

 Membership of editorial boards

 Applications for national and international grants

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 Leadership

 Organisation of conferences

 Training of post-docs and students at master and doctoral level

 Teaching of undergraduates

The grades are given according to the scale presented schematically below.

Excellent = 5

Research at the international forefront. Original research of international interest and published in internationally leading journals. High productivity.

Very good = 4

Research with a high degree of originality but nonetheless falling short of the highest standards of excellence. A good publication profile with mainly publications in internationally leading journals. High productivity and relevance to international research within its subfield.

Good = 3

Research at a good international level with publications in internationally and nationally recognised journals. Research of relevance both to national and international research developments within its subfield.

Fair = 2

Research that only partly meets good international standards. A modest international publication profile.

Limited contribution to research.

Weak = 1

Research of insufficient quality and low productivity. Few international publications. Little original research and little relevance to the field.

As the grade descriptions reflect, the most important criterion was the quality of publications together with the level of productivity. In practice, the various grades used are not as clear-cut as presented above.

Where a group is heavily burdened by other obligations, such as teaching or administration, the Committee has tried to take such circumstances into consideration. When different criteria suggest different grades, a compromise grade was set. In some cases mixed grades are given. Specifically:

 If two grades are separated by a slash (/), this indicates mixed grades within the group; e.g. 4/2 indicates some parts of the group activity are given grade 4, other parts grade 2.

 If two grades are separated by a dash (-), this indicates a grade between the two; e.g. 3-4 indicates a grade somewhere between 3 and 4.

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2.6 The Evaluation Committee

The Evaluation Committee (the Committee) consisted of the following experts, whose profiles may be found in Appendix B:

Professor Aline Bonami Laboratoire MAPMO

University of Orléans, France

Professor Joachim Cuntz Mathematical Institute

University of Münster, Germany

Professor Björn Engquist Department of Mathematics University of Texas at Austin

Professor Barbara Gentz Faculty of Mathematics

University of Bielefeld, Germany

Professor Paul Linden

Department of Applied Mathematics and Theoretical Physics University of Cambridge, UK

Professor Cheryl Praeger

School of Mathematics and Statistics University of Western Australia

Professor Holger Rootzén Mathematical Statistics

Chalmers University of Technology, Sweden

Professor Ulrike Tillmann (Chair) Mathematical Institute

University of Oxford, UK

Anne Pearsall (administrative staff member, Mathematical Institute, University of Oxford) was Secretary to the Evaluation Committee.

Terje Strand PhD (Division for Science of the Research Council of Norway) presented the instructions to the Institutions and to the Committee, made all the practical arrangements and facilitated the hearings and meetings in Oslo.

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3 Evaluation at the National Level

3.1 Assessment of Research

3.1.1 Quality of research

Norway has a strong tradition in mathematical research and has produced some world class mathematicians including Hendrik Abel (1802-1829) and Sophus Lie (1842-1899). Since the last evaluation in 2002, the Abel Prize has been established on the occasion of Abel's 200^th birthday. Each year the Abel Prize ceremonies and Abel Lectures, as well as the Abel Symposia and Abel Committee meetings put Norway in the limelight of mathematics around the world. Thanks to the great efforts of the mathematical community, these and other events associated to the Abel Prize have been highly

successful for the last ten years.

The excellent quality of Norwegian mathematicians was already pointed out in the report of the first evaluation committee in 2002. This present evaluation again shows that a majority of the research groups are of an internationally recognised high standard. A bibliometric analysis carried out by NIFU³, a draft of which was available to the Committee, seems to confirm this view.

The Committee however noticed signs of decline at the top end of the research output in some areas.

While this is difficult to quantify, with the exception of the Emmy Noether Lecture no other invited lecture was given at any of the last three International Congresses of Mathematicians (ICM) by a mathematician included in this evaluation. In contrast, the previous evaluation reported three speakers at the ICM in 1998 alone⁴. The ICMs are held every four years and are a landmark event of the subject, especially for the more theoretical parts. There also seems to be a decline in the number of papers published in the very top journals.

3.1.2 Capacity and Age Structure

There are 168 tenured mathematicians included in this report. This presents a 10% increase since 2002, a slightly higher increase than that of the total population of Norway during the same time span. There is now one mathematician at university for every 29000 inhabitants.

A major concern of the 2002 evaluation committee was the problem of renewal. The Committee is pleased to observe that the replacement of retiring professors by younger people has been managed successfully in many cases. Nevertheless, the situation today remains precarious. As will be pointed out in the next chapter, whole research groups are likely to discontinue because of the retirement of the core members. Though it can be an opportunity for a department to build up a new group and change research direction, it is often easier to attract, and more effective to hire, excellent young people into a vibrant and internationally established research group.

The age of the permanent staff is relatively evenly distributed (Table 1). However, for the 41

mathematicians over 60 years old, and who will retire in the next decade, there are only 44 under 35 years old who may replace them. Given that a fair number of post-docs are expected to prefer jobs in industry and teaching, it is anticipated that there will be a severe shortage of suitable home grown candidates for the tenured positions that will become available in the next decade. The reservoir of excellent young research mathematicians educated in Norway is extremely limited.

3 NIFU: Evaluation of Mathematics – Publication and Citation Analysis, Dag W. Aksnes.

4 Note that Denmark, Finland and Sweden each had a speaker at the ICM 1996 and ICM 2010.

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11 Table 1: Age (and gender) distribution of tenured staff and post-docs per university and age interval in number of persons as of 01.01.2011.

3.1.3 Subfields

Before dividing mathematics into subfields and referring to the traditional and convenient divisions

between pure and applied mathematics, mathematical sciences and statistics, it is important to remember that there is an underlying and underpinning unity of mathematics. Synergies between different branches of mathematics appear unexpectedly and lead often to important breakthroughs. Examples of such developments in the recent past include graphical statistics, the use of persistent homology in data analysis, Voevodsky’s homotopy type theory - a visionary mix of topology and logic/computer science, and the fruitful interactions between analysis and mechanics.

Norway is a small country and the Committee recognises that not every subject can be represented. In this context it is noted that number theory, mathematical physics, and probability are areas particularly under-represented. The only group in number theory is about to be discontinued and mathematical physics is represented only in the work of a few individual researchers.

The geography of the country, the long distances and its sparse population clearly put further constraints on any research community. Despite this, there are several examples where the groups at different universities function essentially as one. One prominent such example is the topology group which is spread over the departments in Bergen, Oslo and Trondheim. Summer schools bring algebraists together from the different universities and, in many cases, professor IIs connect research groups within Norway and abroad.

Analysis

During the last 40 years an outstanding area of excellence in Norwegian mathematics was the field of operator algebras. This field is still represented at an excellent level at Oslo and Trondheim, even though the focus of the field has changed. While traditionally it was considered to be a field in analysis, in recent years it has developed into a more interdisciplinary subject within mathematics. It now has close

connections to algebra, topology, geometry, ergodic theory and recently also to number theory. These new developments are well represented in the groups at Oslo and Trondheim.

Classical and complex analysis is represented in all centres in Norway. The largest group is at Trondheim and functions very well, playing a world-leading role in function theory in connection with problems coming from signal processing. Highlights of their research concern sampling and

interpolation. Original methods have been developed to tackle and solve old problems, and these new methods are now considered as central to the development of new mathematical tools for signal and image analysis (with wavelets, compressed sensing, etc.). There is a researcher in Stavanger working on the same theme, and there are also some connections with Bergen.

Institution Total⁽¹⁾ Women Women % ≤ 30 31-35 36-40 41-45 46-50 51-55 56-60 61-65 ≥ 66

Stavanger 12 0% 1 1 1 1 2 1 2 3

Agder⁽²⁾ 12 2 17% 2 2 3 2 2 1

Oslo 65 8 12% 7 8 8 8 5 9 7 11 2

Bergen ⁽³⁾ 35 3 9% 3 5 5 5 6 3 3 1 4

NTNU 58 9 16% 6 8 9 6 6 2 8 6 7

Tromso 20 2 10% 1 2 3 2 2 5 2 2 1

UMB 9 0% 1 1 2 2 2 1

Total 211 24 11% 17 27 27 26 23 25 25 23 18

Women 24 2 3 6 4 3 2 1 3

(1) These numbers exclude any prof II or associate prof II.

(2) One emeritus was submitted but is excluded here.

(3) This includes five part-time people submitted.

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There is also very good activity in pluri-potential theory, which is currently centred at Oslo and which will also be developed at the highest level in Trondheim with the arrival of an international leader in this area.

Classical/harmonic analysis traditionally has a strong interaction with many areas of analysis, from parts of differential geometry to PDE and applications. This kind of interaction is functioning well in Trondheim and in the smaller universities, where analysts tend to develop contacts, mainly with PDE groups. Of course some harmonic or real analysis is also developed within PDE groups. The small group in Tromsø deserves to be mentioned for its work on differential equations and differential geometry.

Algebra & Algebraic Geometry

There is excellent research activity in the areas of Algebra and Algebraic Geometry in Norway, with research groups in Trondheim, Oslo, Bergen, and within the Pure Mathematics group in Tromsø. World- leading research centred around non-commutative algebras and commutative algebra is conducted in Trondheim, a major recent highlight being the work on cluster categories and cluster tilted algebras. A strong research focus is maintained on the representation theory of finite dimensional algebras and related topics, with explorations of connections to other branches of high current interest, such as hereditary and triangulated categories, representation dimension, and Hochschild homology and

cohomology. Homological methods have also been used to study commutative Noetherian local rings, in particular answering a long-standing open question concerning support varieties for modules.

There has been a long tradition in algebraic geometry in Norway, which continues to be a field that is particularly well represented by an internationally visible and diverse group in Oslo, as well as by smaller groups at Bergen, Trondheim and Tromsø. The topics studied range from classical projective geometry, to deformation theory with applications in Calabi-Yau varieties, symplectic manifolds, and moduli spaces.

Theoretical and algorithmic work is done on enumerative algebraic geometry problems. There is additional focus on combinatorial mathematics, arising from work in algebraic geometry (in Oslo), and also from work on error correcting codes and cryptology (in Tromsø).

Strong links are evident between algebraists and algebraic geometers in different locations, especially between the groups in Oslo and Bergen. The four research groups organise annually an international summer school and a national algebra meeting. Some researchers have had a significant involvement in the Centre of Mathematics for Applications (CMA). Others have developed rather natural links with members of topology research groups, in at least one case resulting in a joint publication.

Most groups are successful in attracting and training masters and PhD students, and have strong international collaborations.

Topology and Geometry

Norway has excellent and internationally visible research activity in the broad area of algebraic topology with centres at Oslo, Bergen and Trondheim. Researchers have been very successful in attracting grants from national and international sources. Through this several conferences and a visitor programme have been organised. Collaboration between the three centres is supported by an RCN project, ‘Topology'.

Research topics and highlights include Galois theory for 'brave new rings', a geometric approach to elliptic cohomology via 2-vector bundles, the constructions of a higher topological cyclic homology, and the constructions of a homotopy theory of C*-algebras.

There is an even age distribution amongst the algebraic topologists, and several good PhD students and post-docs have been trained. Unfortunately, some excellent young people have already been lost from research due to the lack of early career positions.

There is also some research undertaken in the area of differential geometry by some broad and strong members of the pure mathematics groups in Tromsø. Other areas in geometry and topology are at the moment less, or not at all, represented in Norway. There are however good links to and overlap in interest with the research groups in algebra and algebraic geometry, which have resulted in joint publications.

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13 An extension of the activities into the study of manifolds could link the existing research in geometry and topology, and build bridges to activities in analysis.

Partial Differential Equations

The research in partial differential equations in Norway is outstanding. The leading groups are very well known and respected internationally. Non-linear hyperbolic conservation laws have been in focus for quite some time but lately other types of partial differential equations have also been studied. Examples are Hamilton-Jacobi, Yang-Mills, Navier-Stokes and Einstein equations. Recently there has also been substantial progress in equations related to geometry and to stochastic differential equations. The contributions consist of a healthy mix of topics, from abstract questions regarding existence to specific properties with importance for computations. There are also good connections to groups in harmonic analysis and real analysis.

The research is often inspired by applications. A good example is the relevance of non-linear hyperbolic conservation laws to oil reservoir modelling. Other sources for research in differential equations have been general fluid mechanics, electro-magnetics, finance, biology and medicine.

The links between the groups in partial differential equations and those in applied and computational mathematics are overall very strong. These two groups are at some universities even joined into one.

There is some activity in partial differential equations that was not covered in this evaluation, the group in homogenization at Narvik being the most prominent example.

Computational Mathematics

The field of computational mathematics was developed relatively early in Norway. Development in numerical methods for ordinary differential equations is a good example that became internationally well established a long time ago. This field has very successfully evolved into the study of geometric

integrators and methods for stochastic differential equations. Numerical analysis of partial differential equations is now a highly prominent field that has been developed hand in hand with the mathematical analysis of these equations. This collaboration is a Norwegian success story. Research in finite element methods and in methods for non-linear conservation laws are good such examples. Analysis of

geometric algorithms and numerical approximation theory are also well developed. The coupling of this computational research in the Norwegian mathematics departments to numerical linear algebra and high performance computing is perhaps not as developed as in other Scandinavian countries.

The mathematics departments now host most of computational mathematics. In the previous evaluation some of the groups were still part of computer science. There are however very important computational groups at SINTEF and Simula research laboratories, which are not covered in this evaluation.

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Mechanics

Mechanics is well represented in the major centres in Norway. There are prominent groups in fluid mechanics in Oslo and Trondheim, and a smaller activity in Bergen. The research includes theoretical, computational and experimental fluid mechanics and covers a range of applications of relevance to the off-shore industry, renewable energy, climate and environment. These groups have impacts on the international scene especially in the off-shore applications. The groups in Oslo and Trondheim are well supported and seem stable. In Bergen there has been a reduction in activities in oceanography and related topics.

Solid mechanics is restricted to a relatively small group in Oslo, which provides an important educational function.

Stochastics

Statistics in Norway has a very extensive network of cooperation with other disciplines. It provides important input to industry, in particular, and to engineering and medical research. Norwegian research also covers a broad range of central areas in theoretical statistics. Highlights include exciting new ideas in computational and spatial statistics, extensive work in time series analysis, and contributions to model choice and model averaging. In particular there is a large, active and internationally well-founded statistical environment in Oslo. In addition to the statistical group at the faculty of Sciences at Oslo University, which is reviewed here, statistics at the Norwegian computing centre and at the medical faculty are also important parts of this environment. The latter two are not included in this review. The research centre SFI², which is partly included here, has a pivotal role for the Oslo environment. NTNU has a broad, active, and successful statistics group.

There is a strong group in stochastic analysis in Oslo and also some interest in stochastic analysis in the Differential Equations and Numerical Analysis Group in Trondheim. However, apart from this, there seems to be very little research in probability theory in the Norwegian universities.

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3.2 Research Environment and General Recommendations

3.2.1 Importance of Mathematics for Industry and Mobility To quote the 2008 OECD report on Mathematics in Industry⁵,

“Industrial innovation is increasingly based on the results and techniques of scientific research. That research, in turn, is both underpinned and driven by mathematics. […] From the point of view of industry, mathematics is an enabling technology. It provides a logically coherent framework and a universal language for the analysis, optimization, and control of industrial processes. Because it is an enabling technology, its contributions are rarely visible in the final product that industry delivers. Nevertheless, the economic impact is real, and many companies – old, as well as new – have achieved a competitive advantage through the judicious use of mathematics. ”

The interest in industrial mathematics started early in Norway, in connection with big national companies.

More recently, mathematicians are involved in many multi-disciplinary projects. They have developed many collaborations with application oriented research institutes such as the Norwegian Research Center for Offshore Wind Technology and the Norwegian Fishery Research Institute amongst others. This answers partially a requirement of the 2002 report: “encouraging collaboration between mathematics and other sciences” and implies a real impact of mathematics in technology. The same report states that

“many interesting opportunities go unexplored”, speaking of links with industry. The Committee is pleased to report that a considerable effort has been made during the last decade by some departments, in particular in CMA, SFI², and at NTNU.

On the other hand, the Committee does not have the feeling that the necessity of closer contacts between the academic and the industrial worlds was taken on board, as it is at the moment in most European countries. While it is mentioned by the department of mathematics of NTNU that one third of their doctors in Mathematics get jobs in private companies, many colleagues see the industrial sector as

“poaching” the mathematics students with high salaries. A much better reaction would consist of trying to promote doctorates in mathematics inside the non-academic world.

Since the last evaluation, and mainly since 2006, there has been a considerable effort in many European countries in view of industrial mathematics. It is in particular valid in Germany with the Fraunhofer Institute, in the UK with the Knowledge Transfer Network for Industrial Mathematics (KTN) and more recently in France with the creation of a new agency last autumn. It has also been the object of two reports of OECD⁶. In 2011, a book called "European Success Stories in Industrial Mathematics" was published by Springer under the auspices of the European Science Foundation and the European Mathematical Society. It contains the description of 132 collaborations between mathematics and industry. The fact that none of them involves Norway can be taken as a significant indicator of the fact that, probably, the Norwegian mathematical community is absent from these discussions at the European level. Yet Norway has its own agency, SINTEF.

Recommendation:

In these international reports, new mechanisms are proposed with the view of fostering collaboration between mathematics and industry: internships for PhD students or researchers, modeling weeks, and so on. All of them may help to foster mobility between the academic world and companies. They may also help promote doctoral diplomas in the non-academic world.

5 http://www.oecd.org/dataoecd/47/1/41019441.pdf

6 http://www.oecd.org/dataoecd/47/1/41019441.pdf and http://www.oecd.org/dataoecd/31/19/42617645.pdf

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3.2.2 The Nature of Mathematical Research

Mathematics is very much a people intensive activity. Any investment is primarily an investment in people, not machines or expensive laboratories.

For mathematicians, discussions and interactions at conferences (or, equally important, the chance encounters on the corridor of one’s own department) are often the source of new ideas and a short cut to solutions. It is very difficult to know from where the next breakthrough will come.

Unlike many other academic disciplines, mathematicians do not normally work in larger research teams.

A common research project can be helpful but is not necessary for a successful research group, as long as there is a common research interest and critical mass. Many collaborations happen between

individuals often at different international institutions.

Recommendation:

All this makes it difficult to steer the research activities from the outside, and it is often the best strategy simply to support those individuals with a good track record of high quality research.

3.2.3 Funding

Though higher than the OECD average, the expenditure on R&D and innovation per capita in Norway is significantly less than that of Sweden, Denmark and Finland. Furthermore it is important to note that R&D investment is low relative to GDP. While in 2008/09, Sweden and the USA invested circa 3.6% and 2.8%

respectively, Norway spent less than 1.9% of its GDP on R&D⁷. The private sector as a whole seems particularly reluctant to invest in R&D⁸.

For funding purposes, Mathematics is part of the Natural Sciences. In the time period 1993-1995 there was a steep decline in R&D expenditure on the Natural Sciences which only recently has recovered back to the level of funding of 1993. Furthermore, in the time period 1985-2007 the funding for Social

Sciences nearly tripled and that of Medicine & Health nearly quadrupled, whilst the funding for the Natural Sciences did not even double. Both Social Sciences and Medicine & Health now receive more funding than the Natural Sciences in Norway⁹.

Governmental research funding in mathematics at universities in Norway has two routes. One is through the RCN and the other is directly through the central funding of universities. Some departments have successfully attracted additional funds from industry and the European Research Council (ERC) with the result that a high proportion of their funding comes from sources other than university funding (Table 2).

This reflects the high quality of the research undertaken at these places. But the Committee also noted a high variance between percentages for the administrative cost for the different universities from 2% to 24%. Different accounting systems are likely to be at the source of these differences.

Recommendation:

The Committee suggests, however, that this variance is investigated further to ensure that gross

inefficiencies are avoided and that mathematics departments receive 'value for money' from their central administrations.

7 Report on Science and Technology, RCN 2011, pages 11 and 21.

8 Report on Science and Technology, RCN 2011, page 23.

9 Figures supplied by the RCN.

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17 Table 2: Expenditure by mathematics departments per university in 2010.

The Committee is concerned about the severe shortage of young research mathematicians educated in Norway. It appears that this is at least partly due to the lack of funding in basic research and more particularly to insufficient financial support for doctoral students and post-docs.

The Committee observed that this lack of funding is seriously obstructing successful research activity in some of the evaluated groups, even in groups with the highest research reputation. The reason seems to lie in the nature of the Norwegian funding system. Generally speaking, it does not seem to offer sufficient support to basic research. It is especially ill-suited for the more theoretical parts of mathematics.

Applications for funding for basic mathematical research often only fit into the FRINAT programme where - due to insufficient resources - the success rate even for good applications is very low.¹⁰

The Committee found that this fact constitutes a serious impediment for maintaining an internationally competitive research level in mathematics at Norwegian universities.

Recommendation:

The Committee urges the RCN to open a new programme to fund investigator-driven basic research.

Such a programme should also allow for projects with a time perspective of more than three years. In that way excellent groups could have a reliable source of funding for doctoral students and post-docs over a longer period of time. This would allow them to attract doctoral students and excellent post-docs much more easily.

3.2.4 Mathematics Education

There seems to be an increase of mathematics in popular culture in Norway, helped for example by the programme SIFFER, which is shown four times a week on national TV, and by national newspaper coverage of events such as the Abel Prize presentations.

In contrast, departments report that it is difficult to recruit students into their Bachelor degrees programmes and that those who come are often under-prepared for university study.

Potential mathematics students have to be first educated and attracted to the field in schools. This is best done through good teaching by knowledgeable and enthusiastic teachers. The state of supply of mathematics teachers in Norway is however alarming. An international study in 2007 found that 73% of all Norwegian mathematics teachers were over 50 years old, and 36% over 60.¹¹

10 The success rate has recently increased to circa 7% from 5%.

11 TIMSS Advanced 2007

Institution Total ⁽¹⁾ % External Funding University Funding % Non-salary University Funding ⁽²⁾

Stavanger 7.618 1% 7.532 2%

Agder 17.527 14% 15.117 3%

Oslo 103.992 34% 68.236 24%

Bergen 58.196 37% 36.524 11%

NTNU 89.796 27% 65.348 19%

Tromso 17.184 17% 14.185 2%

UMB 7.190 11% 6.392 18%

(1) in 1000 NOK

(2) This is the percentage of University funding that is not spent on salaries for scientific and technical personnel (including social cost), other costs, and infra structure. Most of this is spent on administration.

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Many teachers of mathematics have not studied the subject at college or university.¹²

This situation will be difficult to change quickly. The Committee acknowledges the herculean efforts performed by some departments and individuals who work hard to increase the visibility of their subject through outreach programmes, where current students and faculty visit schools, while others are involved with teacher training programmes. Another route to attracting students into mathematics at Bachelor or research level seems to remain unexplored so far:-

Recommendation:

The Committee would like to encourage departments in collaboration with the RCN to experiment with preparatory summer schools for pupils and summer research projects for undergraduates.

Educating under-prepared students adequately, once they are at university, presents a challenge. It is important that students of mathematics see the theoretical foundations of their subject early on so that more specialized knowledge can be built on top of these and students are not delayed in their

advancement.

Recommendation:

The Committee supports departments that hold the view that every student should have encountered foundational mathematics such as basic group theory and the rigorous definition of limits at the latest by the end of their second year, and if there are suitable students that they should be advised of their options to accelerate to a higher level of mathematics as early as practical.

3.2.5 Career Structure

One factor that makes academic careers in Norway very difficult and sometimes nearly impossible is the lack of career development posts that would bridge the gap between a post-doc position and tenure.

Promising post-docs might see themselves forced at the end of their contract to choose between a non- academic job or finding a position abroad because there is no university position open in their field at that moment in Norway.

Conversely, when there are openings, the shortage of excellent young researchers with a proven track- record appears to be severe in some areas. The Committee has become aware of cases where this has led departments to prematurely and overly promote their junior members.

Recommendation:

As a consequence of these observations the Committee strongly urges the RCN to open a new programme offering junior professorships or positions of junior research group leaders. Such positions would typically follow a post-doctoral position, and should involve teaching and research. A typical period of appointment would be four to six years. Some of these positions could be financed jointly by the RCN and the corresponding universities.

In many countries mobility is required at some stage of the career, especially in mathematical communities. It is generally considered that mobility enhances research productivity. It promotes international contacts and topical mobility.

On the other hand, a systematic mobility requirement may divert some candidates from academic careers. An intermediate solution consists of the development of specific tools to encourage mobility.

The lack of mobility between Norwegian universities and universities or research institutions abroad was commented on in the 2002 evaluation report and remains a problem. This is valid for the whole career, from PhD studies to permanent positions.

12 TIMSS 1997: 54% of all mathematics teachers in grade 6.

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19 Recommendation:

A continuous effort should be made to guarantee some mobility in most careers, in particular for

universities in more remote locations. A certain number of possibilities are available, such as the funding of travels abroad during the doctoral or post-doctoral studies, sabbatical leaves and professor II positions.

This should be systematically developed in departments of mathematics.

Presently RCN funding schemes allow graduate students to spend up to 12 months abroad. The Committee feels however that young researchers would benefit most from their period abroad if they were not already bound to a project but were free to immerse themselves completely into new research activities at an institution other than their home one.

Recommendation:

The Committee therefore recommends that funding structures should be established by the RCN which allow for entire doctoral studies and post-docs to be spent abroad.

3.2.6 Statistical Sciences

The 2002 report was concerned about applied mathematics, and in particular statistics. The Committee has come to the conclusion that research in these areas has generally improved. (See 3.1 and in particular 3.1.5.)

Recommendation:

The Committee recommends continuing the up to now successful efforts directed at the entire spectrum of statistics.

In particular it seems important also to support research in fundamental theoretical statistics. This is a long-term investment which is necessary if Norway is to keep its high level in statistics research.

Recommendation:

The Committee advises departments and research councils to consider strengthening research in probability, both as a foundation for statistics and stochastic analysis, and because of its general mathematical interest.

3.2.7 Gender Imbalance.

Norway is in the fortunate position of having several senior woman mathematicians of high visibility and international status, including the Emmy Noether Lecturer at the last ICM in 2010. Nevertheless, women are under-represented amongst the faculty and student population. The numbers in Table 1 are not encouraging. At the moment little more than10% of all professors, associate professors and post-docs are female. The numbers in each age range are statistically too small to have significance. However, there is an apparent decrease in the number of women from 10/53 down to 5/44 in the age-ranges 35-45 and 25-35, and it is clear that with only 5 women mathematicians below the age of 35 the situation is not likely to improve in the future without special efforts.

The Committee appreciates that most departments consider this a problem, and that some departments have taken steps to improve the situation. It supports initiatives such as in Bergen where women are given extra sabbatical time to prepare for promotion. Often women are called upon to serve on committees beyond what could be considered their expected share. In such cases NTNU has offered support by reducing their workload in other areas or by providing secretarial help. In some departments, professor IIs have increased the visibility of women in a particular research area and provide female students with additional role models.

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20

Still, these measures cannot make up for the low number of younger woman mathematicians in the 'pipeline'. In this context, the Committee notes that the shortage of post-doc positions and early career positions leads to an increased sense of job insecurity, which in turn is a key factor for women in particular to turn to other professions.

Mathematics in Norway is missing many opportunities by not being able to attract and support scientific careers for many more women.

Recommendations:

The Committee strongly recommends that mathematics departments and national agencies such as the RCN take this problem seriously and attack it from all angles. Existing practices that help to create a fair and equal work balance for women and that encourage them to advance their careers should be

maintained and applied more universally. Appointment committees for new positions at any level should seek out specifically women applicants and should be encouraged to include these activities in their reports. Universities, possibly in conjunction with funding agencies such as the RCN, should explore the possibility of creating schemes, which have been pioneered in other countries, where departments receive additional funding when women are appointed, or where women are specifically recruited to post- doctoral positions after a break in their careers.

In the long run the greatest effect is likely to come from establishing a clear career structure and creating a greater sense of job stability through increasing the number of post-doctoral and early career

development positions. The Committee once again urges departments, in collaboration with universities and research councils, to improve the job opportunities of young researchers in the mathematical sciences.

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21

4 Evaluation of Departments and Institutes

4.1 University of Stavanger - Department of Mathematics and Natural Sciences

Mathematics is one of four sections within the Department of Mathematical and Natural Sciences. The Section of Mathematics comprises three research groups: Analysis, Applied Mathematics and Statistics.

There is a heavy teaching commitment within the Mathematics section as it provides a teaching service both to the engineering departments within the faculty and in financial mathematics to the Faculty of Social Sciences. Funding for the department comes almost entirely from the university.

4.1.1 Analysis Grade 2-3

This group consists of 3 professors and 2 associate professors. The main recent activity concerns pluri- complex analysis and harmonic analysis. Three members of the group are approaching retirement. The two recently recruited members engage in a research area of current interest and are very active with a reasonable number of publications in good international journals.

Recommendations

It is necessary to improve scientific joint activity to be able to attract PhD students or post-docs, possibly in connection with the two other groups. More networking should be encouraged; in particular the collaboration with the Analysis group in Trondheim should be strengthened to compensate for the relative isolation of this small group.

4.1.2 Applied Mathematics Grade 3-4

This group consists of 2 professors and 1 associate professor plus 1 externally funded professor.

The research of the group is in the fields of mathematical physics, flow in porous media, and elastic wave propagation. The mathematical physics research is mainly in geometrical aspects of relativity. The modelling of flow in porous media is focused on two-phase flow with applications to oil reservoir

modelling. The latter is natural for Stavanger with its industrial base in the oil industry. The group has a robust publication record with papers in leading journals. There is, however, a lack of grant funding. The research is individually based rather than a group effort. Contacts outside of Stavanger somewhat offset the lack of local collaboration.

Recommendations

The new PhD programme in maths/physics creates opportunities that should not be missed. It will be crucial to generate funding for the programme and proactive efforts to collaborate with other departments and outside groups, in order to get critical mass in grant proposals, are encouraged. A potential

replacement of one retiree should be used to increase the cohesiveness of the group.

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4.1.3 Statistics Grade 3-4

With 1 professor, 1 associate professor, 1 professor II, and 1 externally funded post-doc, the group is very active and carrying out good research. The majority of publications are in medical statistics in collaboration with several university hospitals. The group is well connected within the University of Stavanger. Two students in other departments are co-supervised.

The professor II in the group visits regularly and teaches one course. It is seen very positively that the group was able to organise these regular visits which allow them to profit from the expertise of the professor II.

Recommendations

To provide the group with sufficient external expertise, the opportunity of having a professor II position should be retained. In addition, the group should aim at establishing further international contacts, possibly through sabbatical leaves.

The kind of applied research done within this group may provide funding opportunities. The Committee encourages this group to search actively for such opportunities alongside their present collaboration with university hospitals. This might be in the form of PhD positions or in the form of joint grant applications with researchers from the application area.

4.1.4 Overall Assessments and Recommendations

To address the problem of low student enrolment in Mathematics, the study programme has recently been reorganised, and a new Mathematics and Physics programme has been established at the undergraduate level. At the time of writing the self-evaluations, there were no PhD students within the Section; however a masters programme in mathematics and physics has recently been approved to commence in 2013, along with a PhD programme commencing in June 2011. The Committee is pleased to see this effort at offering a study programme from undergraduate to PhD level.

Recommendations

There seems to be no strategic plan in place for the future of the Section of Mathematics. The Committee advises the section to develop a plan jointly which should deal with future recruitment and questions of group structure. For example the Committee was impressed by the quality of research in Applied Mathematics and encourages the Applied Mathematics and Analysis groups to work together more closely and to consider merging the two groups.

The Committee was surprised by the group structure of the department, which does not correspond with any direction or funding of research activities. The department is strongly encouraged to consider directing efforts in interdisciplinary research towards petroleum engineering to take advantage of local opportunities. In particular the statistics group, which so far is mainly focussing on medical statistics, could benefit from broadening their focus in this way and thus play a more central role at university level.

As a consequence, the department could hopefully attract more funding, external as well as from the university, and become more attractive for students.

In a smaller university sabbaticals are particularly important to give researchers an opportunity to keep up with frontline research. The Committee strongly recommends that the university continues to run a generous programme for sabbatical leave to help keep the University in Stavanger an attractive employer for active researchers.

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4.2 University of Agder - Department of Mathematical Sciences

The Department of Mathematical Sciences is one of four within the Faculty of Engineering and Science at the University of Agder, which is a product of the merging of Adger College and Kristiansand Teacher College in 1994. The department, which comprises 16 professors and associate professors, is mainly oriented towards mathematics education, and delivers master and PhD degrees in mathematics education, while studies in mathematics at Agder College stop at the bachelor level.

Included in this evaluation is the research group in Mathematics. The groups in Mathematics Education and Teacher Education are not in included.

4.2.1 Mathematics Grade 2

The Mathematics group of the University of Agder consists of 6 professors, 5 associate professors and 2 professors emeritus, who are still active in research. Of these, three belong to the Department of Engineering, which is on the other campus of the university, and work in applied mathematics, while the others are in the Department of Mathematical Sciences. The group covers a diversity of fields, including history of mathematics, which is an option in the graduate programme within the Mathematics Education group. Conditions for research within the department are not easy: the teaching load is heavy (the mean percentage of time for research is 25 %). Also, research in mathematics cannot develop without teaching or training at the graduate level. On the other hand, the University of Agder is eager to support research in mathematics, which provides an opportunity to improve research in the mathematics group. This is planned in connection with other groups within the university.

The Mathematics group is very heterogeneous. This concerns both the mathematical areas under study and the quality of research. The small group in functional analysis has good research output, as well as an isolated researcher in fluid mechanics. The activity in history of mathematics is of very good quality, with international recognition. The two professors of the Department of Engineering have a very good scientific output; one in control theory and one on the use of special functions in fluid dynamics and aerodynamics.

4.2.2 Overall Assessments and Recommendations

The Committee appreciates that Agder University gives importance to the quality of teaching in

mathematics and is interested in having researchers in mathematics involved in the training of teachers in mathematics. The opportunity given to improve research in mathematics at the University of Agder should be seized. It seems reasonable to do this in connection with other groups within the university.

Plans for applied mathematics that include colleagues from the Department of Engineering are in the right direction. The possibility of having professor II positions to strengthen links to other more established research centres should be explored.

It is too early to be thinking of a PhD programme in mathematics, but training at the master level should be made possible within the next few years. Available positions should in general be open to all subject areas and advertised in order to attract the best possible candidates.

Present activity in mathematics at Agder University, even though of good quality, is too fragmented to be the sole basis for research in the department in the future. Possible collaboration within the university should be systematically explored.

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4.3 University of Oslo - Department of Mathematics

The Department of Mathematics is part of the Faculty of Mathematics and Natural Sciences at the University of Oslo. The department was founded in 1947 and today it is built up around three sections:

Mathematics, Mechanics and Statistics. Each of the sections is divided into two or more research groups.

Since 2003 the department has hosted a Centre of Excellence called the CMA (Centre of Mathematics for Applications) and, since 2007, has participated in the Centre for Research-Based Innovation called SFI² (Statistics for Innovation). It is currently involved in an application for a further Centre of Excellence for mathematics and statistics in the life sciences in collaboration with the Institute of Basic Medical Research and the Department of Biology.

Over the last 3 years on average 36% of expenditure was externally funded.

4.3.1 Algebra and Algebraic Geometry Grade 5/3/1

This group comprises 3 professors, 2 associate professors, 1 post-doc plus 1 professor emeritus and 1 externally funded staff member. There are currently 6 PhD students within the group, 3 of which are externally funded. There has been 1 PhD graduation in the last 3 years along with 11 MSc graduations.

The group works on a variety of different problems in algebraic geometry. Excellent work is done in projective algebraic geometry with international impact. There is also very good recent work on Calabi- Yau varieties, symplectic manifolds, and enumerative algebraic geometry. The group is very successful in attracting and training masters and PhD students, and it also has strong international collaborations.

Some members have had a significant involvement in the Centre of Mathematics for Applications, and it has presented a challenge for the research group in terms of research focus and maintaining joint activity within the group. In addition, a heavy administrative commitment has severely restricted research opportunity for one member.

Recommendations

Two recent post-doctoral positions have provided an important opportunity for renewal of this research group, since the last permanent appointment was in 1992. However there is a need to consolidate the group and strengthen its internal collaboration by developing a common research programme. The group has an important networking role among other algebra and algebraic geometry groups in Norway,

especially in Bergen and Tromsø. It is strongly encouraged to strengthen these links.

4.3.2 Several Complex Variables Grade 4

The group consists of 1 professor, 1 associate professor, 1 post-doc and 1 professor emeritus. There are currently no graduate students, and only 1 MSc student has graduated in the last 3 years. However, since 2006 the group has had 3 post-docs.

Research activity is centred on holomorphic approximation, complex geometry and dynamics. The group plans to recruit 2 further post-docs and 1 PhD student to work on a new research project financed by the RCN over the next 5 years.

This is a strong group, with an excellent new recruitment. There is a long tradition in pluri-complex analysis in Norway at the highest level. This is pursued in this group, with connections in other universities, and should be maintained even though the present group is of a sub-critical size.

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25 Recommendations

Given the sub-critical size of the present group, the Committee recommends the creation of a larger group in analysis to include this one.

4.3.3 Geometry and Topology Grade 5

The group comprises 4 professors plus 3 professors emeritus and 1 externally funded staff member. The group currently has 3 PhD students; 2 others have graduated in the last 3 years along with 3 MSc

students.

The main interests of the group lie in algebraic topology and algebraic K-theory with applications to geometric topology, motivic homotopy theory and algebraic geometry. The group is coherent and has overlapping interests with the Algebra & Algebraic Geometry group and, to a lesser extent, with the Operator Algebras group. There are a good number of publications, mainly in very highly regarded journals.

The group took the lead in an RCN Strategic University Programme and is a partner of the national RCN project 'Topology'. The University of Oslo identified the group as one of its 'emerging top-tier research groups'. In addition, its leader led an Outstanding Younger Investigator's (YFF) project. The group enjoys high international visibility and has hosted many post-docs and senior visitors in the past.

Recommendations

Since the last evaluation the group has lost three members through retirement but only one new permanent appointment has been made. The group can clearly attract top researchers from abroad as visitors and as post-docs. It also has excellent scientific and administrative leadership. Further

appointments in this area, broadly defined, should be good for the future development of the group, of pure mathematics in the department, and of the subject area within Norway.

4.3.4 Logic Grade 4

The group has 1 professor and 1 type II professor along with 2 professors emeritus. There are no PhD students in the group at present but there have been 2 PhD and 1 MSc graduations in the last 3 years.

The group's main focus is computability and complexity. The group is small but internally and internationally well connected through seminars with logicians from other departments in Oslo, involvement in Computability in Europe and long visits at foreign institutions. This is the only group in mathematical logic in Norway. At the moment it lacks financial security with one of the two members employed as a professor II.

Recommendations

The Committee encourages the department to find a way to secure the future of the group in the university.

4.3.5 Operator Algebras Grade 4-5

There are 2 professors and 2 associate professors who make up the group, as well as 2 very active professors emeritus. There are currently 3 PhD students, 1 of which is externally funded, and 2 MSc students have graduated in the last 3 years.

Evaluation - Research in Mathematics in Norwegian Universities

Research in Mathematics at Norwegian Universities

An evaluation

Evaluation

Division for Science

Research in Mathematics at Norwegian Universities

An evaluation

To the Research Council of Norway

Contents

1 Findings and Recommendations

2 Introduction

2.1 Mandate and the Review Process

2.2 Participants of the Evaluation

2.3 Key Figures

2.4 Previous Evaluation

2.5 Grading

2.6 The Evaluation Committee

3 Evaluation at the National Level

3.1 Assessment of Research

3.2 Research Environment and General Recommendations

4 Evaluation of Departments and Institutes

4.1 University of Stavanger - Department of Mathematics and Natural Sciences

4.2 University of Agder - Department of Mathematical Sciences

4.3 University of Oslo - Department of Mathematics